Classification of the finite dimensional absolute valued algebras having a non-zero central idempotent or a one-sided unity

An absolute valued algebra is a non-zero real algebra that is equipped with a multiplicative norm. We classify all finite dimensional absolute valued algebras having a non-zero central idempotent or a one-sided unity, up to algebra isomorphism. This completes earlier results of Ramírez Álvarez and Rochdi which, in our self-contained presentation, are recovered from the wider context of composition k-algebras with an LR-bijective idempotent.     

被积函数含有绝对值的积分问题

讨论被积函数中含有绝对值的不定积分、定积分和多重积分等问题.     

Eine Bemerkung zum Tripel-Algorithmus zur Bestimmung der kürzesten Wege in einem Graphen

Summary The best algorithm for finding the shortest paths between every pair of vertices in a complete graph is the triple algorithm. This algorithm can be described algebraically as an operation of a category on the set of real valued finite directed graphs.     

Some New Class of Absolute Valued Algebras with Left Unit

Let A be an absolute valued algebra with left unit. We prove that if A contains a nonzero central element, then A is finite dimensional and is isomorphic to \mathbb R, \mathbb C{\mathbb {R}, \mathbb {C}} or new classes of four and eight–dimensional absolute valued algebras with left unit. This is more general than those results in [2] and [3].     

Semi-algebraic Absolute Valued Algebras with an Involution

Abstract

Let A be an absolute valued algebra. In El-Mallah (El-Mallah,M. L. (1988). Absolute valued algebras with an involution. Arch. Math. 51: 39–49) we proved that,if A is algebraic with an involution,then A is finite dimensional. This result had been generalized in El-Amin et al. (El-Amin,K.,Ramirez,M. I.,Rodriguez,A. (1997). Absolute valued algebraic algebras are finite dimensional. J. Algebra 195:295–307),by showing that the condition “algebraic” is sufficient for A to be finite dimensional. In the present paper we give a generalization of the concept “algebraic”,which will be called “semi-algebraic”,and prove that if A is semi-algebraic with an involution then A is finite dimensional. We give an example of an absolute valued algebra which is semi-algebraic and infinite dimensional. This example shows that the assumption “with an involution” cannot be removed in our result.     

Non-anticipative representations of Banach space valued gaussian processes with respect to brownian motion

Necessary and sufficient conditions are found for the equivalence of the measures associated with (i) a Banach space valued Gaussian process, with mean 0, and (ii) a Bach space valued Brownian motion. The notion of a non-anticipative representation of (i) with respect to (ii) is defined and in the case of equivalence of the measures it is shown that such a representation exists and has an explicit stochastic integral form which is invertible. Theorems of Ershov on absolute continuity of measures associated with diffusion processes are extended to Banach space. Applications to infinite-dimensional filtering are considered.     

Algebraic patching over complete domains

We extend the method of algebraic patching due to Haran-Jarden-Völklein from complete absolute valued fields to complete domains. We apply the extended method to reprove a result of Lefcourt obtained by formal patching — every finite group is regularly realizable over the quotient field of a complete domain.     

On sample path properties of semistable processes

This paper contains three main results: In the first result a correspondence principle between semistable measures on L_p, 1 ≤ p < ∞, and Banach space valued semistable processes is established. In the second result it is shown that the paths of a Banach space valued semistable process belong to L_p with probability zero or one, and necessary and sufficient conditions for the two alternatives to hold are given. In the third result necessary and sufficient conditions are given for almost sure path absolute continuity for certain Banach space valued semistable processes.     

Sur les algèbres absolument valuées qui vérifient l'identité (x, x, x) = 0

Let A denote a prehilbert absolute valued real algebra such that (x, x, x) = 0 for all x ε A; for this algebra we obtain the same results we have previously obtained for the flexible absolute valued algebra. Our main theorem is: A has a finite dimension 1, 2, 4 or 8, and is isotopic to or C. One of the results concerning the isomorphism between A and , C^*, or C shows that if for every two idempotents e₁ and e₂ in , then A is isomorphic to , C^*, or C. The example of infinite dimensional Hilbert absolute valued algebra given by Urbanik and Wright indicates that the assumption, (x, x, x) = 0 for all x ε A, is essential.     

Absolute Valued Algebras with Involution

We study absolute valued algebras with involution, as defined in Urbanik (1961 Urbanik , K. ( 1961 ). Absolute valued algebras with an involution . Fundamenta Math. 49 : 247 – 258 . [Google Scholar]). We prove that these algebras are finite-dimensional whenever they satisfy the identity (x, x ², x) = 0, where (·, ·, ·) means associator. We show that, in dimension different from two, isomorphisms between absolute valued algebras with involution are in fact *-isomorphisms. Finally, we give a classification up to isomorphisms of all finite-dimensional absolute valued algebras with involution. As in the case of a parallel situation considered in Rochdi (2003 Rochdi , A. ( 2003 ). Eight-dimensional real absolute valued algebras with left unit whose automorphism group is trivial . Int. J. Math. Math. Sci. 70 : 4447 – 4454 .[Crossref] , [Google Scholar]), the triviality of the group of automorphisms of such an algebra can happen in dimension 8, and is equivalent to the nonexistence of 4-dimensional subalgebras.     

On absolute valued algebras with involution

Let A be an absolute valued algebra with involution, in the sense of Urbanik [K. Urbanik, Absolute valued algebras with an involution, Fund. Math. 49 (1961) 247-258]. We prove that A is finite-dimensional if and only if the algebra obtained by symmetrizing the product of A is simple, if and only if eA_s = A_s, where e denotes the unique nonzero self-adjoint idempotent of A, and A_s stands for the set of all skew elements of A. We determine the idempotents of A, and show that A is the linear hull of the set of its idempotents if and only if A is equal to either McClay’s algebra [A.A. Albert, A note of correction, Bull. Amer. Math. Soc. 55 (1949) 1191], the para-quaternion algebra, or the para-octonion algebra. We also prove that, if A is infinite-dimensional, then it can be enlarged to an absolute valued algebra with involution having a nonzero idempotent different from the unique nonzero self-adjoint idempotent.     

Absolute Valued Algebras Satisfying ( x 2, x 2, x 2) = 0

Abstract

Let A be an absolute valued algebra. We prove that if A satisfies the identity (x ², x ², x ²) = 0 for all x in A, and contains a central idempotent e, that is ex = xe for all x in A, then A is finite dimensional. This result enables us to prove that if A satisfies (x ², x ², x ²) = 0 and admits an involution then A is finite dimensional. To show that our assumptions on A are essential we recall that in El-Mallah [El-Mallah, M. L. (1988). Absolute valued algebras with an involution. Arch. Math. 51:39–49] it was shown that the existence of a central idempotent in A is not a sufficient condition for A to be finite dimensional; and the example given in El-Mallah [El-Mallah, M. L. (2003). Semi-algebraic absolute valued algebras with an involution. Comm. Algebra 31(7):3135–3141] shows that there exist infinite dimensional semi-algebraic absolute valued algebras satisfying the identity (x ², x ², x ²) = 0.     

About the Schur Index for Ambivalent Groups

An ambivalent group is a finite group all of whose irreducible characters are real valued. By Brauer–Speiser theorem, if G is an ambivalent group, then the absolute Schur index m _Q (χ) = m(χ) ≤2. In this note we shall prove that this property is true also for the derived subgroups of ambivalent groups. Also we will prove that there is a relation between the number of conjugacy classes of 2-regular cyclic subgroups of an ambivalent group and the irreducible characters with absolute Schur index 1.     

Integrable and non-integrable structures in Einstein-Maxwell equations with Abelian isometry group <Emphasis Type="Italic">G</Emphasis><Subscript>2</Subscript>

Using the fact that absolute zero divisors in Jordan pairs become Lie sandwiches of the corresponding Tits–Kantor–Koecher Lie algebras, we prove local nilpotency of the McCrimmon radical of a Jordan system (algebra, triple system, or pair) over an arbitrary ring of scalars. As an application, we show that simple Jordan systems are always nondegenerate.     

Nonmonotone, nonlinear evolution inclusions

In this paper, we consider nonlinear evolution problems, defined on an evolution triple of spaces, driven by a nonmonotone operator, and with a perturbation term which is multivalued. We prove existence theorems for the cases of a convex and of a nonconvex valued perturbation term which is defined on all of T × H or only on T × X with values in H or even in X^* (here X - H - X^* is the evolution triple). Also, we prove the existence of extremal solutions, and for the “monotone” problem we have a strong relaxation theorem. Some examples of nonlinear parabolic problems are presented.     

Genre et genre residuel des corps de fonctions values

Let L be a function field of one variable over a valued field (K,|.|), and (|.|_i), 1is, be distinct absolute values over L extending |.| such that the residue fields ¯Lⁱ are function fields of one variable over the residue field ¯K of (K,|.|). We define the defect of the valued function fields (L,|.|_i)/(K,|.|) and prove an inequality between the genus of L/K and that's of ¯Lⁱ/¯K which takes into account the defect, the ramification index of (L,|.|_i)/(K,|.|) and the constant field of Lⁱ/¯K. Our inequality is better than Mathieu's inequality in discretely valued case.     

Numerical Analysis of the Stochastic Moving Boundary Problem

We extend and generalize some recent results on complete convergence (cf. Hu, Moricz, and Taylor [14], Gut [11], Wang, Bhaskara Rao, and Yang [26], Kuczmaszewska and Szynal [17], and Sung [23]) for arrays of rowwise independent Banach space valued random elements. In the main result, no assumptions are made concerning the existence of expected values or absolute moments of the random elements and no assumptions are made concerning the geometry of the underlying Banach space. Some well-known results from the literature are obtained easily as corollaries. The corresponding convergence rates are also established     

Summability and vector amarts

We show that an operator is absolutely summing if and only if it maps amarts into uniform amarts, from which we can deduce a theorem of A. Bellow and another of Edgar-Sucheston. We also show that the absolute value of a Banach lattice valued potential is a potential if and only if the lattice is an A-M space from which we deduce that the L¹-bounded amarts form a Ricsz space if and only if the space is finite dimensional.     

The axiom of choice and almost continuity

F.B. Jones used the Axiom of Choice to define a real valued function of one variable, whose graph intersects every closed subset of the plane that projects onto an uncountable set in the first coordinate [11]. More recently it has been shown that such a function is almost continuous [12]. We use this function to construct a connected space with a dispersion point. And we show that Jones' construction is of central importance in the theory of noncontinuous retractions, which preserve the fixed point property. In particular, we use Jones' construction to characterize the absolute quasi retracts.